The concept of FGMs was first introduced in Japan in 1984 during a space plane project (Niino et al., 1987; Niino et al., 1988a; Niino et al., 1988b; Hirano et al., 1990). The goal was to develop materials that could serve as thermal barriers capable of withstanding extreme temperature gradients. Specifically, the materials needed to endure surface temperatures of up to 2000 K and a temperature gradient of 1000 K across a 10 mm section. This initial application highlighted the potential of FGMs in aerospace and other high-temperature environments. Early research on FGMs focused on the inverse optimum design of the phase composition utilizing classical or empirical micro-structure models and the development of processing techniques to create the desired gradation in the phase composition. Figure 1a depicts the concept of FGMs with ceramics, metal and micro-pore (Hirano et al., 1991), Figure 1b depicts the phase distributions of two different materials (V_A and V_B) with micro-pore (V_P) expressed by power function V_A(x) = 1 – V_B(x)ⁿ–V_P(x), Figure 1c shows Eshelby’s Ellipsoidal Inclusion model (Eshelby, 1957). Table 1 shows several Rules of Mixtures based on Micro-structures (Hirano et al., 1990).

Researchers explored various methods including chemical vapor deposition (Niino et al., 1988a; Sasaki et al., 1989), powder metallurgy (Niino et al., 1988b; Sata, 1993) and centrifugal casting (Fukui, 1991) to fabricate FGMs with tailored properties. As the field progressed, the scope of FGMs applications expanded beyond aerospace to include biomedical, automotive, and energy sectors. For example, FGMs have been used in dental implants (Kondo et al., 2004) and bone prostheses due to their ability to mimic the natural gradation of properties found in biological tissues (Bohuder et al., 2014).

As mentioned above, the micro-structure modeling and optimum design of the phase composition in FGMs played a crucial role in the development of FGMs. Early models were primarily based on linear analyses, focusing on simple gradation profiles and basic material properties. However, with advancements in computational power and numerical methods, more sophisticated models emerged. In recent years, the integration of advanced computational techniques, such as Finite Element Method (FEM) (Kawai et al., 1992; Teraki et al., 1992; Arai et al., 1993) and multi-scale modeling, has further enhanced the ability to design and optimize FGMs. These methods enable researchers to predict the behavior of FGMs to tailor their properties under various loading conditions including fabrication and specific applications. The historical development of FGMs reflects a continuous effort to improve material performance through innovative design and modeling techniques. The advancements in micro-structure modeling have been instrumental in realizing the full potential of FGMs across a wide range of applications.

Subsequently in Japan, fundamental research on Functionally Gradient (Graded) Energy Conversion Materials (EC-FGMs) was advanced as a national project to dramatically improve the conversion efficiency of energy conversion materials (Hirano et al., 1994). Thermoelectric (TE) conversion materials and photovoltaic conversion materials, among other energy conversion materials, are inorganic semiconductor materials. To improve their properties, precise prediction of electronic properties and control of the micro-structure of the materials are necessary. The micro-structure refers to all crystal defects (lattice defects), including grain boundaries, pores, precipitates, dislocations, and stacking faults. Figure 2 is a conceptual diagram of the structure and organization of ceramics introduced by Yasui, (1982) and applied to micro-structures of FGMs (Hirano, 1995). In the figure, the micro-structures are classified by physical scales. Actual materials consist of complex combinations of these microscopic structures, and the physical properties at the crystal lattice level rarely appear directly as macro properties. Therefore, in material design, theoretical considerations must be made at each physical scale of the microscopic structures of the material, namely, the crystal lattice level, the micro-structure level, and the solid material/device as a whole.

For example, TE conversion devices operate under large temperature gradients to generate electricity via the Seebeck effect. Therefore, in addition to evaluating the power generation efficiency of TE materials, it is essentially multi-physics to assess and mitigate thermal stress distributions at the interfaces between electrodes and TE elements, as well as within composite TE materials.

Also, to evaluate the performance of TE devices, it is essential to solve multi-physics phenomena that couple thermal and electrical transport processes. Figure 3 shows the analysis model of TE EC-FGM module.

And the set of coupled differential equations derived from the Boltzmann transport equation are shown in Equation 1 (Hirano, 1995), where j denotes the electric current density, q the heat flux, ф the electric potential, T the temperature, σthe electrical conductivity, α the Seebeck coefficient, and κ the thermal conductivity (lattice κ _L + electronic κ _e ). j q = σ T α σ σ α k + T σ α 2 − ∇ φ − ∇ T (1)

FEM allows for the simulation of complex geometries and material behaviors by discretizing the material into smaller elements. In the initial phase of FGMs development, inverse design procedures based on analytical solutions under steady-state conditions were employed, particularly for infinite plates and axisymmetric cylindrical geometries, to optimize the graded compositional distribution (Hirano et al., 1990). However, in order to validate these designs against experimental evaluations of synthesized FGMs, unsteady thermal conduction and thermal stress analyses were subsequently performed using two-dimensional finite element models that incorporate edge effects (Kawai et al., 1992; Teraki et al., 1992). Also, the thermo-elasto-plastic behavior of FGMs under thermal loading and the evaluation of residual stresses after unloading were evaluated (Arai et al., 1993). Since then, FEM has become a widely adopted tool for the analysis and design of FGMs.

Suresh and Mortensen published a book of comprehensive review on FGMs (Suresh and Mortensen, 1998), which includes the fundamentals of processing and thermomechanical response of graded metals and metal/ceramic composites, covering broad range of topics including process metallurgy, composite synthesis, the mechanics and micromechanics of composites, and fracture mechanics. Moreover, it includes one chapter on FEM describing fundamentals of FEM for FGMs, thermal stress analysis, boundary conditions and modeling, stress-strain behavior and fracture mechanics, and practical applications and validations (Suresh and Mortensen, 1998). On the other hand, Reddy published a paper on the development of FEM elements tailored for multilayered plates with graded compositions (Reddy, 2000), which has significantly enhanced the accuracy of FGM modeling and evaluation along with the integration of advanced FEM techniques such as adaptive meshing and multi-physics simulations.

Multi-scale modeling is a powerful approach that integrates information across different length scales, from the microscale to the macroscopic level. This method allows for a comprehensive understanding of how micro-structural features influence the overall properties of FGMs. While Homogenization Method (Bendsoe and Kikuchi, 1998) was developed for modeling periodic micro-structures and widely implemented in commercial FEM solvers such as Ansys and ABAQUS, a different approach has been proposed for thermal stress-relaxation type FGMs (Rhee, 2007). In this approach, the geometric characteristics of the microcells are explicitly parameterized within each finite element, and the graded microscale problems are solved using a non-periodic homogenization method. Then, the results were incorporated into the macroscale FEM stiffness matrix (Rhee, 2007).

As mentioned above, micro-mechanics analyses focused on understanding the relationship between the micro-structural features of FGMs and their macroscopic properties. On the other hand, a theoretical investigation based on fracture mechanics was conducted by Erdogan for the first time to evaluate the fracture strength of FGMs in the presence of initial micro-cracks introduced during fabrication (Erdogan and Wu, 1997). In this study, a plane elasticity problem was formulated for FGMs in which the Young’s modulus varies continuously in the thickness direction. Surface-perpendicular cracks were introduced to model both edge and embedded cracks. Under three distinct loading conditions, the stress intensity factors (SIF) were theoretically derived. Furthermore, the crack opening displacement (COD) and stress distributions were analyzed to clarify the influence of FGMs gradation on fracture behavior (Erdogan and Wu, 1997). This work has provided a theoretical foundation for the fracture-safe design of FGMs. Furthermore, a toughness enhancement design utilizing stress-induced phase transformation based on micro-mechanical analysis was also proposed for metal–ceramic FGMs (Tukamoto, 2017).

The effect of material property variation on ductility and fracture strain in FGMs was investigated using FEM and the Gurson-Tvergaard-Needleman (GTN) model (Shahzamanian et al., 2020), which is strain-controlled for void nucleating. The material properties of FGMs in the tensile tests are represented by a power low distribution in the thickness direction. The ductility and fracture strain changes with gradation index. The development of stress triaxiality slows down in FGMs, and the total void volume fraction is reduced (Shahzamanian et al., 2020). Figure 4 shows the comparison of thermal stress distributions in conventional thermal barrier coating and FGMs (Shahzamanian et al., 2020).

To support the design and lifetime prediction of thermal barrier coatings (TBCs) using FGMs in gas turbine blades and high-temperature structural components, thermal fracture behavior analyses focusing on structures with pre-existing edge cracks and multiple internal cracks arranged to imitate a curved interface have been reported recently (Petrova and Schmauder, 2021). Another new method to evaluate fracture mechanism of FGM was also introduced recently, which is based on extended FEM (XFEM) developed by Belytschko (Moes et al., 1999). It is reported that XFEM can analyze crack propagation without remeshing, and capture crack paths influenced by property gradation, ensuring high accuracy in Stress Intensity Factor (Sinha and Pamnani, 2025).

Another application of FGMs was recently reported, in which polymeric laminated composite materials including polymer FGMs are investigated numerically by FEM for improving the tribological behavior and experimentally verified (Nowler et al., 2025). By designing a gradual variation in material properties across the laminate thickness, the authors demonstrate reduced wear rates, lower friction coefficients, and improved stress distribution. The study includes FEM modeling and tribological testing, showing that FGM-based laminates outperform conventional ones under sliding contact conditions (Nowler et al., 2025).

As explained in historical background, fundamental research on the Energy Conversion FGMs (EC-FGMs) was started to improve the conversion efficiency of energy conversion materials (Hirano et al., 1994; Hirano, 1995). In this situation, precise estimation of thermal and electronic transport properties and tailoring of micro-structures were required. For the estimation of the transport properties, different scales of physical structures, ranging from lattice level crystal structure, grain level micro-structure and macro scale device structure are considered. Several estimation methods of the material properties for the design of EC-FGMs were considered. As one of the energy conversion materials, thermoelectric (TE) materials were investigated, and models were presented for the transport properties of heavily doped TE semiconductors (Hirano et al., 1994; Hirano, 1995; Hicks and Dresselhaus, 1993; Whitlow and Hirano, 1995; Nishio and Hirano, 1997; Chen et al., 2003). The dimensionless figure-of-merit (ZT) plays a decisive role of TE materials as the critical measure of the energy conversion performance. Equation (2) defines ZT by several material parameters, where α, σ, κ _L , κ _e , and T are the Seebeck coefficient, electrical conductivity, lattice thermal conductivity, electronic thermal conductivity, and absolute temperature, respectively. Z T = α 2 σ κ L + κ e T (2)

To increase ZT the use of quantum well was proposed to increase the product α ²σ by Hicks and Dresselhaus (Hicks and Dresselhaus, 1993), which introduces quantum confinement effects that modify the electronic density of state leading to an increase in the Seebeck coefficient and a reduction in thermal conductivity.

Thereafter, the carrier energy filtering concept applied to Super Lattice structure was firstly proposed in the EC-FGMs project (Whitlow and Hirano, 1995). The concept was improved more precisely to include ionized impurity (electron and hole) carrier energy filtering and phonon (acoustic and optical) scattering into the theoretical approach for the optimization of the parameters α ²σ, κ _L andκ _e influencing the charge‐carrier energy spectrum and phonon scattering, through structural changes or the material and the scattering interactions (Nishio and Hirano, 1997). The theoretical calculation results were shown that phonon scattering effects are quite dominant rather than ionized impurity scattering (Nishio and Hirano, 1997).

Chen and Dresselhaus reported a comprehensive review on the advances in the field of TE materials (Chen et al., 2003). They focused on two major areas as shown below and proposed so-called Phonon Engineering, which aims to decrease the lattice thermal conductivityκ _L by means of phonon scattering in addition to reduce the electronic thermal conductivityκ _e due to electron scattering (Chen et al., 2003).

AI and Deep Learning (DL) have become powerful tools for the modeling of micro-structures. For example, a novel design framework was constructed by integrating two types of neural networks. Deep Convolutional-Generative Adversarial Network (DC-GAN), which generates candidate micro-structures that satisfy geometric constraints, and Convolutional Neural Network (CNN), which serves as a surrogate model to learn the relationship between microstructural geometry and its corresponding physical properties (e.g., compliance tensor) (Tan et al., 2020). By combining these networks into a unified “design network,” inverse design becomes feasible, enabling the efficient generation of micro-structures that exhibit desired material properties (Tan et al., 2020). Also, application of DL to inverse design of phase separation structure in polymer alloy was reported (Hiraide et al., 2021), in which the micro-structures are dependent on the volume fraction of polymer A and the product of the repulsive interaction and chain length as shown in Figure 5a). In this study, forward process to estimate Young’s modulus from micro-structure image utilized CNN, and inverse design process was done by GAN as shown in Figure 5b) (Hiraide et al., 2021).

Another machine learning framework was proposed that emulates the thinking process of metallurgists, with the aim of identifying microstructural features that critically influence material properties (Noguchi et al., 2022). Usual machine learning approaches often lack interpretability due to insufficient understanding of the underlying physical mechanisms and causal relationships. In this work, by mimicking the intuitive design process of human experts, the Vector Quantized Variational Auto-Encoder (VQ-VAE) seeks to automatically extract key structural features without relying on explicit physical models, and PixelCNN enables highly accurate structure generation and reconstruction as shown in (Figure 6) (Noguchi et al., 2022). The target material system is artificially designed dual-phase steel, and the optimization focuses on fracture-related properties (Noguchi et al., 2022). The same authors also presented on Baysian inverse inference of material properties from micro-structure images (Noguchi and Inoue, 2024).

A novel framework for predictive micro-structure image generation was presented (Azqadan et al., 2023) utilizing Denoising Diffusion Probabilistic Models (DDPMs) (Ho et al., 2020). By leveraging the iterative denoising process inherent to DDPMs, the proposed method enables the synthesis of high-fidelity micro-structure images conditioned on material-specific parameters such as composition, processing history, and target properties (Azqadan et al., 2023). The generated images exhibit statistically consistent features with experimentally observed micro-structures, including grain morphology and phase distribution. This approach offers a promising solution to the inverse design problem in materials science, facilitating the generation of plausible micro-structures from desired macroscopic properties (Azqadan et al., 2023). Figure 7 shows the whole processes of proposed system with DDPM.

More comprehensive survey for AI-driven approaches for multi-scale modeling was reported recently (Peng et al., 2025), which includes parameterized micro-structures and FEM homogenization with meshes based on diffuse interface representation and DL based surrogate model training. With the trained surrogate models, micro-structure optimization or inverse design of micro-structure can be efficiently executed (Peng et al., 2025). It also introduced MCRpy (Seibert et al., 2022), an open-source Python framework designed for the characterization and reconstruction of micro-structures in materials science displayed in Figure 8. Micro-structures, such as grain boundaries, pores and phases, play a critical role in determining the macroscopic properties of materials. MCRpy provides a modular and extensible platform for analyzing micro-structural features using statistical descriptors and for reconstructing synthetic micro-structures that match desired characteristics. The framework supports integration with machine learning models and optimization algorithms, enabling data-driven materials design and analysis (Seibert et al., 2022).

The Transformer is a deep learning architecture introduced by Vaswani et al., in 2017 (Vaswani et al., 2017), which has since become foundational in a wide range of machine learning applications, particularly in natural language processing (LLM), computer vision (ViT), and scientific domains such as Materials Informatics (most famous example is AlphaFold (Jumper et al., 2021)). The Transformer architecture is composed of two primary modules: the encoder and the decoder. The encoder processes the input sequence and generates a contextualized representation, while the decoder utilizes this representation to generate the output sequence. A key innovation of the Transformer is the self-attention mechanism, which allows the model to dynamically weigh the importance of different elements in the input sequence. This mechanism is further enhanced by multi-head attention, which enables the model to attend to information from multiple representation subspaces simultaneously. To compensate for the lack of inherent sequential structure, positional encoding is added to the input embeddings, allowing the model to incorporate order information (Vaswani et al., 2017).

An innovative approach to the design of architected materials directly from natural language descriptions was proposed (Yang and Buehler, 2021). The authors employed a transformer-based neural network architecture, combining CLIP and VQGAN, to generate material structure images from textual prompts (e.g., “a regular lattice structure of steel”), which were subsequently used to construct 3D models and to fabricate them via 3D printing realizing a “Words to Matter” pipeline (Yang and Buehler, 2021). This approach introduced a novel design paradigm that differs fundamentally from conventional Computer Aided Design (CAD) or numerical design methods by enabling intuitive linguistic expressions to be directly translated into material architectures (Yang and Buehler, 2021).

A hybrid algorithm with Transformer and CNN encoders, called as CS-Net, for micro-structure image segmentation in materials science was proposed (Alrfou et al., 2023), which also utilized Transfer Learning for models pre-trained specifically on microscopic images achieved higher segmentation accuracy as compared to those trained on natural image datasets (Alrfou et al., 2023). Figure 9 shows the Encoder-Decoder architecture and the test data.

The first large-scale benchmark on generative design of inorganic materials using transformer-based language models was presented (Fu et al., 2023), including GPT, BART, and RoBERTa Language models, that have demonstrated success in natural language processing and molecular design, are applied to the generation of inorganic material compositions, and their performance is evaluated in terms of chemical validity, novelty, and property control. The authors trained and compared seven transformer models using composition data extracted from large-scale materials databases such as ICSD, OQMD, and the Materials Project (Fu et al., 2023). Figure 10 shows the candidate new structures discovered by MT-GPT2 model with zero e-above hull (Fu et al., 2023).

A Transformer-based language model developed for the prediction of polymer properties was proposed recently (Xu et al., 2023). Traditional approaches to evaluating polymer characteristics often rely on costly and time-consuming experiments or simulations. This study introduces a novel methodology that leverages the Transformer architecture to directly predict polymer properties from their sequences. The model employs a chemically-informed tokenizer to represent polymer structures and is pretrained using masked language modeling (MLM) on a large corpus of unlabeled polymer sequences. TransPolymer demonstrates superior performance across multiple downstream property prediction tasks. Figure 11 shows the key architecture of TransPolymer (Xu et al., 2023).